On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction

TL;DR

This paper investigates the failure of a refined BMV conjecture's upper bound for matrix traces, explains the reasons behind this failure, and proposes a corrected pinching refinement that offers a sharper structural decomposition.

Contribution

It identifies the limitations of the original conjecture, explains the failure mechanism via spectral bridges, and introduces a new pinching-based inequality with proven cases.

Findings

Counterexamples show the original upper bound fails for 3x3 matrices.

The failure is linked to off-diagonal spectral bridges and non-canonical common parts.

The proposed pinching refinement provides a sharper, proven inequality in specific cases.

Abstract

We analyze why the refined Bessis--Moussa--Villani conjecture fails. The refined conjecture proposed that the normalized trace average over all words with prescribed numbers of letters \(A\) and \(B\) should be bounded above by the clustered word \(\Tr(A^nB^m)\). Recent counterexamples of Cha show that this upper bound is false already for \(3\times3\) positive semidefinite matrices when \(n=m=5\). We explain the failure from the viewpoint of commutative common parts. The term \(\Tr(A^nB^m)\) is not the canonical common part of the pair \((A,B)\); it is only one clustered word. After pinching \(B\) relative to \(A\), the natural commuting contribution is \(\A_{n,m}(A,\EA(B))\). The off-diagonal complement \(B-\EA(B)\) creates spectral bridges, and mixed words can distribute the powers of \(A\) along closed cycles more efficiently than the clustered word. This gives a mechanism for finding counterexamples. Motivated by this mechanism, we propose a corrected pinching refinement \[ \A_{n,m}(A,B)\ge \A_{n,m}(A,\EA(B)). \] We prove this corrected conjecture in the case of two letters \(B\), obtaining a sandwich refinement \[ \A_{n,2}(A,\EA(B)) \le \A_{n,2}(A,B) \le \Tr(A^nB^2). \] Thus, even where the old clustered upper bound remains true, the pinching viewpoint gives a sharper structural decomposition.